Arslan F.

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Genus Calculations of Complete IntersectionsFeza Arslan and Sinan SertözBilkent UniversityDepartment of MathematicsCommunications in Algebra, 26(8) (1998), 2463-2471.1 IntroductionIn this article we calculate the genus of a projective complete intersectionof any dimension and the genus of an affine complete intersection curve ofsuperelliptic type. The arithmetic genus of a projective complete intersectiondepends on the degrees of the hypersurfaces cutting it. A formula for its genususing only the degrees of these hypersurfaces is given by Hirzebruch in [3] asan application of his Riemann-Roch theorem. Miranda has communicatedus the following algorithm [5]:One has to compute the cohomology H i(OX(k)) for every complete inter-section X ⊂ P n, for every twist k, and for every cohomology group H i. Startwith X = P n itself, where all is known. Assume you know everything when Yis a complete intersection with degrees (d1, d2, ..., dr) and let X be a completeintersection with degrees (d1, d2, ..., dr, dr+1). Hence X is the intersection ofY with a hypersurface of degree dr+1. Then use the exact sequence0 −→ OY (k − dr+1) −→ OY (k) −→ OX(k) −→ 0.Knowing the cohomology on the left and the middle will tell you the coho-mology on the right. Applying this recursive algorithm with X a curve givesthe genus computation as g = h1(OX).We calculate the Hilbert polynomial of a projective complete intersectionby using the Koszul resolution. The arithmetic genus is then obtained fromthe constant term of Hilbert polynomial.In the affine case the usual procedure for finding the genus of a curve isto find the genus of a resolution of its projective closure. This is intractablein general. We restrict our attention to the types of curves which are used incoding theory. These are superelliptic curves which we describe in equation(5). Stepanov and Özbudak has calculated the genus of some elliptic typecurves in [9] using the “counting the differentials” method (for an accountof this, see [7]). Stepanov also calculates the genus of an important classof curves in [8], see Remark 1 after Theorem (4) at the end of this arti-cle. Özbudak has also calculated the genera of fibre product of superellipticcurves using the “counting the differentials” and the Riemann-Hurwitz genusformula [6].Khovanskii has given a formula for the arithmetic genus of an algebraicvariety X defined in (C − 0)n by a nondegenerate system of polynomialequations f1 = ... = fk = 0 with polyhedra ∆1, ...,∆k. His method dependson choosing a toral nonsingular compactification Mn of (C − 0)n and usingan exact sequence of sheaves and cohomology groups of Mn [4].Our method involves describing the projective closure. After describingthe ramification indices contributed by resolving the singular points at in-finity, a direct application of the Riemann-Hurwitz formula yields the genusas a function of the degrees of the hypersurfaces cutting the curve. We alsoshow how to count the regular differential on the curve.Our main results are Theorem (1) and Corollary (2) for the projectivecase and Theorem (4) and Corollary (5) for the affine cases.We thank R. Miranda, A. Klyachko and S. Stepanov for several discus-sions during the preparation of this work. The first author also would liketo thank W. Vasconcelos, TÜBİTAK and Rutgers University Department ofMathematics for their generous support and hospitality during his visit.2 Projective CaseLet S denote the homogeneous coordinate ring, k[x0, ..., xn], of Pnk , wherek is an algebraically closed field, usually C. We assume that there arehypersurfaces H1, ..., Hr of Pnk of degrees d1, ..., dr respectively such thatXr = H1 ∩ · · · ∩ Hr. The hypersurfaces H1, ..., Hr correspond to homoge-neous polynomials f1, ..., fr ∈ S of degrees d1, ..., dr respectively.22.1 The Hilbert Polynomial of XrTheorem 1 The Hilbert polynomial Hr(z) of Xr is given by the followingformulaHr(z) = ϕ(z) +r∑m=1(−1)m∑1≤i1<···<im≤rϕ(z − di1 − · · · − dim) (1)whereϕ(z) =1n!(z + 1)(z + 2) · · · (z + n) =(z + nn).Proof: From [1, Theorem 2], the Koszul complexK(f1, ..., fr) (which is theKoszul complex of the homomorphism ϕ : Sr = ⊕ri=1Sei → S = k[x0, ..., xn]such that ϕ(ei) = fi for 1 ≤ i ≤ r) is a free resolution of S/(f1, ..., fr).Namely, we have the following exact sequence0→ ∧r(Sr)→ ...→ ∧2(Sr)→ Sr → S → S/(f1, ..., fr)→ 0. (2)In [2], in order to grade∧m(Sr) =⊕1≤i1<...<im≤rSei1 ∧ ... ∧ eim (1 ≤ m ≤ r), (3)a degree di1 + ...+ dim is assigned to a basis element ei1 ∧ ...∧ eim , so that (2)is an exact sequence with maps homogeneous of degree zero. Now imposingthe additive property of Hilbert polynomials on the exact sequence (2), theformula given in (1) is obtained. 2Corollary 2 The arithmetic genus, ga(Xr), of Xr is given by the formulaga(Xr) =r∑m=1(−1)m+n−r∑1≤i1<···<im≤rϕ(−di1 − · · · − dim). (4)33 Affine CaseIn this section, we compute the genus of a complete intersection curve C inAn+1Cgiven by,yd11 = (x− a11)...(x− a1m)yd22 = (x− a21)...(x− a2m)......ydnn = (x− an1)...(x− anm)(5)where 2 ≤ d1 ≤ · · · ≤ dn ≤ m− 1 and all aij’s are distinct, with aij ∈ C.This is a smooth affine curve and its projective closure C in Pn+1Cissingular. Let C̃ be a resolution of C. The genus of C is then defined as thegenus of C̃. In the following subsections we will in turn describe the projectiveclosure of C, describe a finite map from C̃ to P1, count the ramification indicesof the points of C̃ under this map and finally apply the Riemann-Hurwitzformula to this map to calculate the genus.3.1 A Finite Morphism to P1In order to compute the genus of a nonsingular model C̃ of the projectiveclosure C of C we first define a finite morphism from C̃ to P1.There exists a finite morphismϕ : C → C(x, y1, ..., yn) 7→ xC is embedded into Pn+1 the same way C embeds into P1. The morphism ϕextends to C algebraically by definingϕ : C → P1[x : y1 : · · · : yn : 1] 7→ [x : 1][0 : y1 : · · · : yn : 0] 7→ [1 : 0]See also the parametrization (8) for a justification of this definition. If C̃ is aresolution of C, then C and C̃ are isomorphic everywhere except at finitelymany points which correspond to the points at infinity and ϕ extends overto C̃ by sending all the points at infinity to [1 : 0] as above.4Thus we have a mapϕ : C̃ → P1which is a morphism of degree d1d2...dn.3.2 Ramifications of ϕWe first examine the n = 2 case with d = d1 = d2. Consider the curve C1given by the equationsyd1 = xm + a1xm−1 + ...+ am−1x+ am =: F1(x)yd2 = xm + b1xm−1 + ...+ bm−1x+ bm =: F2(x)(6)with 2 ≤ d ≤ m − 1. For the points in the affine plane we can take x as alocal parameter. When x is not equal to any of the aij’s then the ramificationof ϕ at x is 1. When x = aij, then the ramification of ϕ at x is d. (For thegeneral case of equation (5) the ramification at aij is d1 · · · d̂i · · · dn, where d̂idenotes that the term should be omitted.)To examine the points at infinity choose a local parameter t with x = 1/t.Then we haveyd1 = xm + a1xm−1 + · · ·+ am−1x+ am= (1/t)m + a1(1/t)m−1 + · · ·+ am−1(1/t) + am= (1 + a1t+ · · ·+ am−1tm−1 + amtm)/tm.Letd = acm = bc, (a, b) = 1, c ≥ 1. (7)Define a new local parameter T such thatT a = t.Then the above parametrization of yd1 becomesyac1 = (1 + a1Ta + · · ·+ amT abc)/T abc.5Similarly we haveyac2 = (1 + b1Ta + · · ·+ bmT abc)/T abc.Let H1(T ) and H2(T ) be power series such that yac1 = Hac1 (Ta)/T abc andyac2 = Hac2 (Ta)/T abc. Then the points around infinity are parametrized asP (α1, α2, T ) =[T bα1T aH1(T a): 1 :α2α1H2(Ta)H1(T a):T bα1H1(T a)], (8)where α1 and α2 are d-th roots of unity. Note that H1(0) = H2(0) = 1 andthus the points at infinity are of the form [0 : 1 : α2/α1 : 0]. In the T -planelet T1 and T2 be two points such that T2 = λT1 where λ is an a-th root ofunity. We have T a1 = Ta2 but Tb1 6= T b2 since (a, b) = 1. Hence P (α1, α2, T1) 6=P (α1, α2, T2). As T ranges in the T -plane P (α1, α2, T ) describes a branch ofthe curve at infinity. There are then d2/a = dc branches at infinity. Sincethere are d points at infinity, around each such point there are then c branchesmaking the total of dc branches. Each branch corresponds to a different pointon the resolution so there are dc points on the resolution corresponding to thepoints at infinity, i.e. the cardinality of the set ϕ−1([1 : 0]) ⊂ C̃ is dc. Totalramification index for the preimage of any point under ϕ, i.e. the degree ofϕ, is d2. This gives a ramification index of a for each point in the resolutioncorresponding to the point at infinity.In the general case when d = d1 = · · · = dn, the total ramification indexof ϕ is dn, there are dn−1c branches at infinity each having ramification indexa. This is the case for the curve define with the equations (9).In the most general case, see equations (5), when d = d1 = · · · = ds <· · · < dn there are ds−1c branches at infinity each with ramification indexads+1 · · · dn. In this case the cardinality of ϕ−1([aij : 1]) is d1 · · · d̂i · · · dn andthe ramification index of each such point is di − 1. The total degree of ϕ isdsds+1 · · · dn.Before we apply the Riemann-Hurwitz formula to find the genus we sum-marize our observations about the points at infinity of the curve C of equation(5) in the following corollary.Corollary 3 Let C be the curve in An+1 defined by (5), with d1 ≤ d2 ≤. . . ≤ dn. Assume that for some s the first s di’s are equal, i.e. d = d1 =6d2 = ... = ds < ds+1 < ... < dn. Then the projective closure C of the curve Cin Pn+1 is the union of C and the points of the form,[x : y1 : · · · : ys : ys+1 : · · · : yn : z] = [0 : 1 : α2 : · · · : αs : 0 : · · · : 0]where αd2 = ... = αds = 1.23.3 The Genus CalculationThe Riemann-Hurwitz formula for the map ϕ takes the formgC = 1− degϕ+12∑x∈C(ex − 1)= 1− degϕ+ 12∑x∈ϕ−1([∗:1])(ex − 1) +12∑x∈ϕ−1([1:0])(ex − 1),where ex denotes the ramification index.Theorem 4 Let C be the complete intersection curve given by,yd1 = (x− a11)...(x− a1m)yd2 = (x− a21)...(x− a2m): :ydn = (x− an1)...(x− anm)(9)where d + 1 ≤ m, and all aij’s are distinct. The genus of C is given by theformulagC = 1−12(d−mnd+mn+ c) dn−1 (10)where c = (d,m).Proof: The degree of ϕ is dn. The ramification index at smooth finitepoints is 1 and for each point x ∈ C for which ϕ(x) = aij the ramification7index is d. There are dn−1 points in ϕ(x) = aij and the number of aij’s ismn. This gives 12mndn−1(d− 1) for the first summation in (10).There are dn−1c points on the resolution of the projective closure of Ccorresponding to points at infinity. Each such point has ramification indexa. This then gives 12dn−1c(a− 1) for the second summation in (10). Puttingthese in and simplifying gives the seeked formula. 2Remark 1: Putting in d = 2, c = 1 we recover Stepanov’s formula1 + (mn− 3)2n−2, see [8, p370, Lemma 1]. Stepanov arrives at this formulaby constructing an explicit basis for the differential forms of the curve. Heworks over a finite field Fq of characteristic p > 2.We finally give the formula for the most general case.Corollary 5 Let C be the complete intersection curve given by (5), withd1 ≤ d2 ≤ . . . ≤ dn and with the first s di’s equal to d. The genus of C isgiven by the formulagC = 1−12(d−mnd+ms) ds−1ds+1 · · · dn −12ds−1c−mds2n∑i=s+1ds+1 · · · d̂i · · · dn, (11)where c = (d,m).2Remark 2: This corollary can be proved in the same way as Theorem(4). The ramification values required for the formula are given at the end ofsection (3.2).Remark 3: Note that when we put s = n in the above formula (11) werecover the formula (10) of Theorem (4). However this is only an algebraicphenomena since geometrically the two formulas are derived from differentconfigurations at infinity.83.4 Counting the DifferentialsIt is of interest to summarize the method of counting the regular differentialsfor the curve C of equation (9). See also the equations (7) for the conventionsin use. For any point in the affine space let x be a local parameter andconsider the regular 1-formω(j1,...,jσ)i1,...,iσ =dxyj1i1 · · · yjσiσwhere 1 ≤ σ ≤ n, 1 ≤ i1 < · · · < iσ ≤ n and 1 ≤ j1, ..., jσ ≤ d − 1. Bychecking the order of vanishings of x and yi’s it can be shown that the formω(j1,...,jσ)i1,...,iσ is regular at any point in the affine space. Let x∞ be any point atinfinity on the projective closure of C. Let ν∞ denote the order of vanishingof a function at x∞. Choosing t = 1/x as a local parameter around x∞ weobserve thatν∞(x) = −aν∞(yi) = −b.Let ω̄(j1,...,jσ)i1,...,iσ denote the expression for ω(j1,...,jσ)i1,...,iσ around x∞. We then haveν∞(ω̄(j1,...,jσ)i1,...,iσ ) = (j1 + · · ·+ jσ)b− a− 1and if P (x) is a polynomial then P (x)ω(j1,...,jσ)i1,...,iσ is regular at x∞ if and onlyif degP (x) ≤ ((j1 + · · · + jσ)b − a − 1)/a. We can then give a basis for theregular differential 1-forms;{xrω(j1,...,jσ)i1,...,iσ | σ = 1, ..., n, 1 ≤ i1 < · · · < iσ ≤ n,1 ≤ j1, ..., jσ ≤ d− 1,0 ≤ r ≤ ((j1 + · · ·+ jσ)b− a− 1)/a }.The cardinality of this set then gives the genus of the curve C. It turns outthat the required formula isg(C) =n∑σ=1∑1≤i1<···<iσ≤nd−1∑j1=1· · ·d−1∑jσ=1|| (j1+···+jσ)b−1a||, (12)9where || || denotes the greatest integer function. Note that this formula nowworks on any algebraically closed field of any characteristic, when a 6= 0.Stepanov has calculated this sum for d = 2 and c = 1 over a field ofcharacteristic p > 2, [8, p372], (in that case d = a = 2 and m = b is odd).The sceptic reader can check the validity of equation (12) using a com-puter; as equation (10) suggests the genus grows exponentially as n increasesso we suggest an example with a small n, for example n = 3. Let d = 66,m = 385. Then a = 6, b = 35 and c = 11. Putting these in equations (10)and (12) both gives the genus as 163 345 645.References[1] Eagon, J. A., Northcott, D. G., Ideals defined by matrices and a certaincomplex associated with them, Proc. Roy. Soc. (A), 269 (1962), 188-204[2] Eagon, J. A., Northcott, D. G., A Note on the Hilbert functions ofcertain ideals which are defined by matrices, Mathematika, 9 (1962),118-126[3] Hirzebruch, F., Topological Methods in Algebraic Geometry, Springer-Verlag, 1978.[4] Khovanskii, A. G., Newton polyhedra and the genus of complete inter-sections, Funkts. Anal. Prilozhen., No. 1 (1978), 51-61[5] Miranda, R., Private communication.[6] Özbudak, F., Codes on Fibre Products of Some Kummer Coverings, toappear in Finite Fields and Their Applications.[7] Shafarevich, I., R., Basic Algebraic Geometry I, Springer-Verlag, 1977[8] Stepanov, S., Character sums and coding theory, Finite Fields and Ap-plications, (1996), 355-378.[9] Stepanov, S. A., Özbudak, F.,Fibre Products of Hyperelliptic Curves andGeometric Goppa Codes, Discrete Math. Appl., 7 (1997), 223-229.sarslan@fen.bilkent.edu.trsertoz@fen.bilkent.edu.tr10

Genus calculations of complete intersections

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Genus calculations of complete intersections

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